Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph primitives. For $m$-edge and $n$-vertex graphs, it is well-known to be solvable in $O(m\sqrt{n})$ time; however, for several applications this running time is still too slow. We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More specifically, we focus on (almost) linear-time kernelization. We start a deeper and systematic study both for general graphs and for bipartite graphs. Our data reduction algorithms easily comply (in form of preprocessing) with every solution strategy (exact, approximate, heuristic), thus making them attractive in various settings.

中文翻译：

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用于最大匹配的线性时间数据缩减的威力

在无向图中寻找最大基数匹配可以说是最核心的图基元之一。对于 $m$-edge 和 $n$-vertex 图，众所周知在 $O(m\sqrt{n})$ 时间内可解；然而，对于一些应用程序来说，这个运行时间仍然太慢。我们研究了线性时间（和几乎线性时间）数据减少（用作预处理）如何缓解这种情况。更具体地说，我们专注于（几乎）线性时间内核化。我们开始对一般图和二部图进行更深入和系统的研究。我们的数据缩减算法很容易（以预处理的形式）符合每个解决方案策略（精确、近似、启发式），从而使它们在各种环境中都具有吸引力。